String Theory and M-Theory

9.8 Nonperturbative effects in Calabi–Yau compactifications 403
Let us use the same SU(3) covariant notation as in the previous exercise.
Compactification on a Calabi–Yau three-fold again gives N = 2, D = 4
supersymmetry. The fields belong to the following supermultiplets:
gravity multiplet : Gµν, Ψµ, Ψµ, Cµijk
h 2,1 vector supermultiplets : C µij ¯ k , Gij, fermions
h 1,1 hypermultiplets : C µνi¯j, G i¯j, B i¯j, C i¯j, fermions
universal hypermultiplet : Φ, C, Bµν, Cµν, fermions.
Now taking into account that Gij is complex and that the four-form C has a
self-duality constraint on its field strength, the total number of the massless
scalar fields is 2h 2,1 + 4(h 1,1 + 1). The total number of massless vector fields
is h 2,1 + 1. ✷
9.8 Nonperturbative effects in Calabi–Yau compactifications
Until now we have discussed perturbative aspects of Calabi–Yau compactification
that were understood prior to the second superstring revolution.
This section and the following ones discuss some nonperturbative aspects
of Calabi–Yau compactifications that were discovered during and after the
second superstring revolution.
The conifold singularity
In addition to their nonuniqueness, one of the main problems with Calabi–
Yau compactifications is that their moduli spaces contain singularities, that
is, points in which the classical description breaks down. By analyzing a
particular example of such a singularity, the conifold singularity, it became
clear that the classical low-energy effective action description breaks down.
Nonperturbative effects due to branes wrapping vanishing (or degenerating)
cycles have to be taken into account.
To be concrete, let us consider the type IIB theory compactified on a
Calabi–Yau three-fold. As we have seen in the previous section, the moduli
space M2,1 (M) can be described in terms of homogeneous special coordinates
XI . A conifold singularity appears when one of the coordinates, say
X 1
= Ω, (9.140)
vanishes. The period of Ω over A 1 goes to zero, and therefore A 1 is called
A 1